Related to all this is a point raised in
the last of Professor V. S. Ramachandran's
wonderful Reith Lectures in 2003. On hearing the lectures,
I was moved to post here an
`Einsteinian
footnote'
to what was said in the lecture about the ancient problems of `self',
`consciousness', and `free will', and the possibility of a
solution `staring at us all along'...
Basic to it all, though often overlooked,
is combinatorial largeness. This
includes the unimaginably large
number of ways for complex systems to go wrong, a point
familiar to computer programmers.

I've moved my older discussions of those matters to two separate pages.
The first is
a little factsheet on CO2
-- a few facts about CO2
that attract no serious controversy.
The second touches on
some of the wider implications,
trying to bring out the distinction between
the climate-system amplifier's input variables or `control knobs'
(such as anthropogenic CO2) and
its internal or feedback variables
(such as water vapour, and naturally-fluctuating CO2).
The discussion emphasizes what we know from studies of past climate,
independently of the imperfections of the big climate models.

Note added December 2013: I just came across a recent book
with well-documented insights into why there's been so
much confusion about CO2 and climate:
Merchants of Doubt: How a Handful of Scientists Obscured the
Truth on Issues from Tobacco Smoke to Global Warming,
by Naomi Oreskes and Erik M. Conway, Bloomsbury, 2010.

Here's a brief essay
`On
thinking probabilistically' (pdf, 0.2 Mbyte).,
based on the beautiful
theorems of Richard Threlkeld Cox.
It is a reprint from the 2007 proceedings of
the 15th 'Aha Huliko'a Workshop on Extreme Events
held at the University of Hawaii in January 2007.
It tries to address some of the most deep-seated difficulties
in understanding
probability and statistics and, by implication, in understanding
science itself.
Even more than usual, the difficulties stem from unconscious assumptions.
I try to show how all this is related to natural selection and
why there's far more to it than
the outdated `frequentist versus Bayesian' polemics.

Frequentist thought-experiments are very useful
in some circumstances -- an important working tool.
But there is, I believe, a serious problem with
the old `hardcore frequentism' and its
influence on the teaching of undergraduates.
The trouble begins with the tacit
portrayal of probabilities as absolutes
-- as the probability of this or that
(i.e., with conditioning statements suppressed).
I believe this teaching practice to be deeply confusing,
and sometimes very dangerous,
as with the notorious cases of unsafe murder convictions via the
prosecutor's fallacy and even simpler statistical fallacies
(as in the
Sally
Clark case,
the probability that she didn't kill her babies,
etc). But the deepest and most dangerous confusion of all
comes from the hardcore frequentist or absolutist
view of probability values as properties of things in the
outside world, or material world --
i.e., as properties of
what science calls objective reality.

The `walking lights' display at the top of this page reminds us of
how we perceive reality, namely by unconsciously
fitting internal mental models to data.
Data consist of information arriving
from the outside world, such as patterns of light on
the retinas of our eyes.
Science works in fundamentally the same way,
though more slowly and more consciously and with more and better data.
So a coherent account of what science is requires us
not only to assume that reality exists but also,
crucially, to maintain a clear distinction between reality,
on the one hand, and models of it on the other. Models --
theories if you will -- are
partial and approximate representations of reality, some models
being better than others.
Probability theory is one of the most powerful tools at our disposal
for building good models of reality.
Indeed, it's arguably an indispensable tool for that purpose
(e.g., p. 158 and footnote 5 of the
essay;
see also the
literature on countless scientific topics including
quantum theory, statistical mechanics, noisy dynamical systems,
`stochastic parametrization' and stochastic modelling in general).
So, in any coherent account of what science is and how it works,
probability values and probability distribution functions
need to be regarded as model properties,
alongside all the other mathematical constructs we use in model-building.

So to insist that probabilities are, on the contrary,
properties of things in the real material world
is to preclude a clear understanding of what science is.
We cannot distinguish between models and
reality if the distinction is hopelessly blurred
at the outset. And such confusion is incalculably dangerous.
That's no exaggeration
in a world whose fate depends on a clear understanding of
science, and on the wise use of science.
Here's
a conference talk
that pursues these points a bit further
(pdf, 1.2 Mbyte),
first given on 26 September 2007.
(Of course there's no original thought here -- the clarifying ideas go
back to Plato, Kant, Laplace and R. T. Cox
and have been well vindicated by
experimental psychology in recent decades, including systematic
and detailed studies of the walking-lights phenomenon.)

The wise use of science in the UK, indeed the survival of good
science, is still under threat from our outmoded libel
laws especially, at present, in Scotland and Northern Ireland.
There has been progress in other parts of the UK. For recent news
take a look at http://www.senseaboutscience.org.uk/
My own take on how Big Money threatens good science is in
the Prelude and Postlude to my new
e-book under construction.

Here is
Rupert Ford's last published paper,
on imbalance and inertia-gravity-wave radiation
and written jointly with Warwick
Norton and myself.
A Rupert Ford Memorial Fund has been established;
for more information go to
this page
where also,
by kind courtesy of Professor E. David Ford,
Rupert's remarkable PhD thesis is now available as a
searchable pdf.

The related articles for the Encyclopedia of
Atmospheric Sciences are
here.

Here is the latest
on air-sea interaction
(fundamental fluid dynamics of wind-generated water waves).

To see preprints of the McIntyre-Norton and Ford-McI-Norton papers
on potential-vorticity inversion and on the slow quasimanifold
and
Lighthill radiation
(which came out in the Millennium May Day issue of J. Atmos. Sci.),
click
here. There is a small but important
CORRIGENDUM here, also in J. Atmos. Sci.58, 949,
15 April 2001.
The original 1996 report
with Roulstone on velocity splitting in Hamiltonian balanced models is
here.
The
review with Roulstone, in press for CUP and incorporating the
tutorial material from the 1996 report
(plus various updates and
a primer in Kähler and hyper-Kähler geometry)
is
now available here;
and preprints are still available on request.
Also shortly available will be
a preprint of the work with Mohebalhojeh on
non-Hamiltonian velocity splitting,
and a recent conference paper
(Limerick Symposium)
that tries to summarize our present knowledge of
balance and potential-vorticity inversion
and some still-outstanding mysteries.
This last link also leads to a beautiful
animated version of Figure 3 of the conference paper,
displaying
CRISTA data,
by kind courtesy of Dr Martin Riese of the University at Wuppertal.

A few reprints are still available, on request, of my review chapters
for Meteorology at the Millennium
(Academic Press and Royal Meteorological Society)
and for Perspectives in Fluid Dynamics
(Cambridge University Press), on the fluid-dynamical fundamentals of
large scale atmospheric circulations -- anti-friction and all that,
now out in paperback.
The Meteorology chapter was written more specifically for an
atmospheric-science audience; in addition, it
reviews the recent progress
in understanding the
solar tachocline
in the light of today's knowledge of
terrestrial stratospheric dynamics.

To find the
polar cooling thought-experiment,
click here (2.8K). &nbsp
This is in section 6 of the review
`Atmospheric dynamics: some fundamentals, with observational
implications' written for the
Proceedings of the International School of Physics
`Enrico Fermi',
CXV Course, 1993.

For my anonymous ftp site (which has been mirrored on the web server)
click
here.
It holds mostly miscellaneous preprints, corrigenda
and reprints, including the `airsea' files (new ideas about wind-generated
water waves), and material for a book in preparation on
lucidity
and science (3.6K), related to the animation above. Comments welcome!
NB: some of the files
are compressed into the old Unix .Z format. These are
recognized, and can be uncompressed, by the standard utility
gunzip.

For the Campaign for Science and Engineering
(formerly Save British Science),
click
here,
and for related matters
here (7K)
and
here (5K).
The last two links point respectively to the celebrated Halloween Documents
and Eric S. Raymond's book The Cathedral and the Bazaar.
Between them they illustrate
why survival of the spirit of open science will continue to be socially and
commercially important, and how great will be our peril if we forget this.
It is this same spirit of open science, with its remarkable ideal and ethic
-- whose problem-solving power was discovered only a few centuries ago,
in Renaissance times -- that has made possible an astonishing achievement of
recent times: the development of complex yet reliable software, reliable
enough for vast systems like the Internet to function. The
Halloween Documents
testify to this in an unexpectedly cogent way.

Living organisms are more complex still. The
Halloween Documents and related commentaries --
including the story of how the entire
Internet nearly came under the control of a single giant corporation,
in a parallel to World War Two
--
have given us reason to hope
that the spirit, ideal, and ethic of open science will sooner or later
be recognized in the commercial, as well as in the academic, world
as a prerequisite to the safety and reliability of -- for instance --
genetic engineering. Such recognition might help to turn the tide of
madness in, for instance,
patent law,
arguably a major cause of technological hazard.
See the important new book by Sulston and Ferry
referenced there. This also gives us
an
insider's view of the human genome project.

Back to the workaday present.
Here's a link to my draft-revision toolkit,
lucidity-supplem.txt
(2.7K). Mainly for colleagues and students.

Here's
the web version of my lecture notes for
the Maths Methods III NST class on small oscillations and
group theory, including representation theory and character tables.
NST stands for the Cambridge Natural Sciences Tripos.
The notes (now with a logical slip on page 40 corrected) can be
downloaded as a pdf file (ca. 0.5Mbyte).
Here is the
first examples sheet for 2008,
and here is
the second.
Note that there's a solution to sheet 2 q6 embedded in the
lecture notes, about halfway down page 80.

(Yours truly at a Cambridge street party on 5 June 2012,
photo courtesy of Owen Spencer-Thomas.)

Research interests

My own research interests, and those of the Atmospheric Dynamics
research group at Cambridge,
have been mainly oriented toward understanding the fluid dynamics
of the Earth's atmosphere, with emphasis on the stratosphere, at
altitudes between about 10 and 50 kilometres, which contains most of
the ozone shield. Our interests extend also to other parts of the
atmosphere and to the oceans, whose fluid dynamics is in many ways
fundamentally similar. Recently I have been working on some
astrophysical problems as well, to which our terrestrial
knowledge has contributed key insights. These include the
Sun's differential rotation
and the
formation of jets on Jupiter.

Work in the Atmospheric Dynamics group has
helped to explain, for instance, why
the strongest ozone depletion occurs in the southern hemisphere,
`even though' the chlorofluorocarbons and other chemicals causing it are
emitted mainly in the northern hemisphere. This is a story of the epic
journeys of atoms and molecules, circumnavigating the globe many times
before arriving in the Antarctic polar stratosphere.

Understanding the atmosphere means understanding a nonlinear,
multi-scale, chaotically-evolving fluid motion intimately coupled to
radiative heat transport and chemistry. Data from modern terrestrial
and space-based observing systems tell us a great deal about what
happens; and the challenge is to understand why -- a prerequisite to
predicting what will happen in future.

Some aspects of the problem are already well understood, but many
challenges remain. We try to deploy all the means at our disposal
-- mathematical theory, thinking by analogy, testing ideas with
numerical experiments, comparison with data and, occasionally,
experimentation on a small scale with real fluid-dynamical systems to
which an idea under consideration applies. Something that thrills me
personally is seeing, with the help of an appropriately general
theory, how fluid phenomena you can easily observe in the kitchen
sink [see
The Quasi-Biennial Oscillation...]
can, surprisingly, help to make sense of certain phenomena on
the relatively grand scale of the entire atmosphere -- including
three particular phenomena that used to be counted among the great
enigmas of atmospheric science.

The first is the so-called `quasi-biennial oscillation'
(QBO), observed since the early 1950s in the equatorial lower
stratosphere, when the operational meteorological network became
sufficiently developed. The east-west winds reverse direction
roughly every fourteen months, throughout a belt encircling the globe,
a remarkable example of order out of chaos and long-term predictability
-- and regarding causal mechanisms a total enigma for nearly two
decades, whose solution began to emerge only in the 1960s, when I was
a graduate student. To see a beautiful laboratory analogue of the QBO
(the Plumb-McEwan experiment), including an animated visualization, click
here. (If you want to repeat the experiment, first read
`Inside Stories'.)

The second phenomenon, and one-time enigma,
is that of the extraordinarily low temperatures
observed over the summer pole at altitudes just over 80 kilometres.
Temperatures as low as 105 Kelvin (minus 168 Celsius) have been
observed there -- far lower than anywhere else on, in, or above
the Earth, despite the strong solar radiation incident on the summer
pole. (Simple geometry shows this solar radiation to be stronger, in
diurnal average, than anywhere else on Earth.)

The third phenomenon and, at first sight unrelated, enigma is what
used to be called the turbulent `negative viscosity' due to
large-scale eddies in the subtropical stratosphere and upper
troposphere, and specifically recognized as enigmatic in Edward N.
Lorenz' classic monograph `The Nature and Theory of the General
Circulation of the Atmosphere', published in 1967 by the World
Meteorological Organization in Geneva.

The gyroscopic pumping pulls air gently but persistently upward and
poleward out of the tropical troposphere and lower stratosphere, then
pushes it back downward
toward the extratropical troposphere, the greater
part of it through the winter stratosphere via complicated, chaotic
pathways. The distinction between tropics and extratropics is, for
this purpose,
purely dynamical: the tropics feels the Earth's rotation far less.
Typical large-scale upwelling velocities in the tropical lower
stratosphere (altitudes 15 to 20 km) are seasonally variable roughly
from 0.2mm/s in northern summer to 0.4mm/s in northern winter, or
roughly 6 to 13 km per year, with the largest values confined mainly
to the most intense month or two of the northern winter.
This sets the e-folding timescale
for removal of chlorofluorocarbons from the troposphere,
because rates of land and ocean uptake of chlorofluorocarbons are at
least a decimal order of magnitude slower. This means that it would
take several centuries for chlorofluorocarbon concentrations
to diminish to 1 percent of their present values,
if all sources were somehow turned off tomorrow.
This same `Brewer-Dobson circulation'
plays a large part in determining the rate of
replenishment of stratospheric ozone, of the order of megatonnes
per day.

Wave breaking, understood in a suitably general sense that becomes
apparent from theoretical studies of `wave-mean interaction', plays a
crucial role in the wave-induced angular momentum transport. This in
itself is a major challenge for theoreticians and numerical
modellers. It means for one thing that the atmospheric circulation
cannot be thought of as a simple turbulent fluid, to which classic
turbulence theories and related concepts like Fickian `eddy
diffusivity' or `eddy viscosity' might apply.
Rather, the atmosphere viewed on almost
any scale confronts us with a highly inhomogeneous, multi-scale
`wave-turbulence jigsaw puzzle', in which wavelike and turbulent
regions are often adjacent, and influence each other very strongly,
and in which the net effect can often be `anti-frictional' -- tending
to drive the system away from, not toward, solid rotation.
Progress has depended, and will continue to depend, on clever
combinations of theoretical thinking and computer modelling, all the
way up to high-resolution numerical experiments run on the most
powerful supercomputers. All this is very much part of the
group's ongoing work under Professor Peter Haynes.

I have written a major review of the
fluid dynamical fundamentals, at early
graduate-student level, focusing on the three enigmas and
forming chapter 11, pp.557-624, of a new book Perspectives in Fluid
Dynamics: A Collective Introduction to Current Research edited by
G. K. Batchelor, H. K. Moffatt, and M. G. Worster. It was
published in hardback by Cambridge University Press in November 2000
and in paperback in January 2003.
[As
noted above, please kindly read
each wedge in the equations
as a (non-associative vector-product) cross;
I believe the printed formulae
are otherwise correct. I'd be glad to send a corrected copy to anyone
interested. The paperback edition incorporates these and a few other
corrections.]
The unifying theme is the fluid dynamics of
large scale atmospheric circulations, with a few remarks
on the opposite-extreme case of
the so called thermohaline, or meridional overturning,
circulation (MOC) of the oceans.

For more about the research group's work, especially in more
recent years, see its
publications pages.

Note:&nbsp If you are interested in applying to do PhD work here
then you may want to look at
the relevant administrative information, which is available
here.&nbsp
I'm now retired but
the work of the group continues under
Professor Peter Haynes,
and applications are encouraged from interested people with
good degrees in mathematics or physics. Some further information
is available on this site under
Courses and Opportunities.